We consider the utilization of power series to determine solutions to certain differential equations.
Series Solutions Near an Ordinary Point I
Many physical applications give rise to second order homogeneous linear differential equations of the formwhere , , and are polynomials. Usually the solutions of these equations can’t be expressed in terms of familiar elementary functions. Therefore we’ll consider the problem of representing solutions of (eq:7.2.1) with series.
We assume throughout that , and have no common factors. Then we say that is an ordinary point of (eq:7.2.1) if , or a singular point if . For Legendre’s equation,and are singular points and all other points are ordinary points. For Bessel’s equation, is a singular point and all other points are ordinary points. If is a nonzero constant as in Airy’s equation, then every point is an ordinary point.
Since polynomials are continuous everywhere, and are continuous at any point that isn’t a zero of . Therefore, if is an ordinary point of (eq:7.2.1) and and are arbitrary real numbers, then the initial value problemhas a unique solution on the largest open interval that contains and does not contain any zeros of . To see this, we rewrite the differential equation in (eq:7.2.4) as and apply Theorem thmtype:5.1.1 with and . In this section and the next we consider the problem of representing solutions of (eq:7.2.1) by power series that converge for values of near an ordinary point .
We state the next theorem without proof.
We call (eq:7.2.6) a power series solution in of (eq:7.2.5). We’ll now develop a method for finding power series solutions of (eq:7.2.5). For this purpose we write (eq:7.2.5) as , where
Theorem thmtype:7.2.1 implies that every solution of on can be written as Setting in this series and in the series shows that and . Since every initial value problem (eq:7.2.4) has a unique solution, this means that and can be chosen arbitrarily, and are uniquely determined by them.
To find , we write , , and in powers of , substitute into (eq:7.2.7), and collect the coefficients of like powers of . This yieldswhere are expressed in terms of and the coefficients of , , and , written in powers of . Since (eq:7.2.8) and the first part of Theorem thmtype:7.1.6 imply that if and only if for , all power series solutions in of can be obtained by choosing and arbitrarily and computing , successively so that for . For simplicity, we call the power series obtained this way the power series in for the general solution of , without explicitly identifying the open interval of convergence of the series.
Equations like (eq:7.2.10), (eq:7.2.11), and (eq:7.2.12), which define a given coefficient in the sequence in terms of one or more coefficients with lesser indices are called recurrence relations. When we use a recurrence relation to compute terms of a sequence we’re computing recursively.
In the remainder of this section we consider the problem of finding power series solutions in for equations of the formMany important equations that arise in applications are of this form with , including Legendre’s equation (eq:7.2.2), Airy’s equation (eq:7.2.3), Chebyshev’s equation, and Hermite’s equation, Since in (eq:7.2.16), the point is an ordinary point of (eq:7.2.16), and Theorem thmtype:7.2.1 implies that the solutions of (eq:7.2.16) can be written as power series in that converge on the interval if , or on if . We’ll see that the coefficients in these power series can be obtained by methods similar to the one used in Example example:7.2.1.
To simplify finding the coefficients, we introduce some notation for products: Thus, and We define no matter what the form of .
Computing the coefficients of odd powers of from (eq:7.2.20) yields
The results in Examples example:7.2.1 and example:7.2.2 are consequences of the following general theorem.
In the examples considered so far we were able to obtain closed formulas for coefficients in the power series solutions. In some cases this is impossible, and we must settle for computing a finite number of terms in the series. The next example illustrates this with an initial value problem.
A Note on Technology
Computing coefficients recursively as in Example example:7.2.4 is tedious. We recommend that you do this kind of computation by writing a short program to implement the appropriate recurrence relation on a calculator or computer. You may wish to do this in verifying examples and doing exercises.
If you’re interested in actually using series to compute numerical approximations to solutions of a differential equation, then whether or not there’s a simple closed form for the coefficients is essentially irrelevant. For computational purposes it’s usually more efficient to start with the given coefficients and , compute recursively, and then compute approximate values of the solution from the Taylor polynomial The trick is to decide how to choose so the approximation is sufficiently accurate on the subinterval of the interval of convergence that you’re interested in. In the computational exercises in this and the next two sections, you will often be asked to obtain the solution of a given problem by numerical integration with software of your choice (see the end of Trench 10.1 for a brief discussion of one such method), and to compare the solution obtained in this way with the approximations obtained with for various values of . This is a typical textbook kind of exercise, designed to give you insight into how the accuracy of the approximation behaves as a function of and the interval that you’re working on. In real life, you would choose one or the other of the two methods (numerical integration or series solution). If you choose the method of series solution, then a practical procedure for determining a suitable value of is to continue increasing until the maximum of on the interval of interest is within the margin of error that you’re willing to accept.
In doing computational problems that call for numerical solution of differential equations you should choose the most accurate numerical integration procedure your software supports, and experiment with the step size until you’re confident that the numerical results are sufficiently accurate for the problem at hand.
Trench, William F., ”Elementary Differential Equations” (2013). Faculty Authored and Edited Books & CDs. 8. (CC-BY-NC-SA)